The Cantor–Schröder–Bernstein Theorem for ∞-groupoids

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We show that the Cantor–Schröder–Bernstein Theorem for homotopy types, or ∞-groupoids, holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent. The argument is developed in the language of homotopy type theory, or Voevodsky’s univalent foundations (HoTT/UF), and requires classical logic. It follows that the theorem holds in any boolean ∞-topos.


Original languageEnglish
JournalJournal of Homotopy and Related Structures
Early online date28 Jun 2021
Publication statusE-pub ahead of print - 28 Jun 2021


  • Cantor–Schröder–Bernstein Theorem, ∞-groupoid, Homotopy type theory, Univalent foundations, ∞-topos

ASJC Scopus subject areas